AI 中文总结
本文证明对任意Coxeter群，源于Richardson簇仿射铺砌的移位下Bruhat区间[e,w]·x⁻¹是EL-可壳化的，还明确了其分级结构及唯一极大、极小元的构成。
AI 中文摘要
设W为任意Coxeter群，移位Bruhat区间[w₁,w₂]·x⁻¹是Bruhat区间[w₁,w₂]经元素x平移得到的集合，其偏序由W的Bruhat序定义。这些偏序集源于Richardson簇的仿射铺砌，且一般既非扭曲区间也非倾斜Bruhat区间。本文主要结果为：对任意Coxeter群，移位下区间[e,w]·x⁻¹是EL-可壳化的，证明方式是通过用反射对每个覆盖进行显式标记。同时，本文还证明[e,w]·x⁻¹是一个分级偏序集，其唯一极大元由Demazure乘积给出，唯一极小元由本文引入的反向Demazure算子给出。
英文摘要
Let $W$ be an arbitrary Coxeter group. The shifted Bruhat interval $[w_1,w_2]\,x^{-1}$, the translate of the Bruhat interval $[w_1,w_2]$ by an element $x$, is partially ordered by the Bruhat order of $W$. These posets arise from affine pavings of Richardson varieties, and in general they are neither twisted intervals nor tilted Bruhat intervals. Our main result is that the shifted lower intervals $[e,w]\,x^{-1}$ are EL-shellable for every Coxeter group, via an explicit labeling of each cover by a reflection. Along the way we show that $[e,w]\,x^{-1}$ is a graded poset with a unique maximum given by the Demazure product and a unique minimum given by an opposite Demazure operator that we introduce.
Comments26 pages, 4 figures
